Model 1

# Summary statistics of CGPA
summary(admission_data$CGPA)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   6.800   8.170   8.610   8.599   9.062   9.920

summary(admission_data$Chance.of.Admit)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.3400  0.6400  0.7300  0.7244  0.8300  0.9700

summary(admission_data[, c("CGPA", "Chance.of.Admit")])

##       CGPA       Chance.of.Admit 
##  Min.   :6.800   Min.   :0.3400  
##  1st Qu.:8.170   1st Qu.:0.6400  
##  Median :8.610   Median :0.7300  
##  Mean   :8.599   Mean   :0.7244  
##  3rd Qu.:9.062   3rd Qu.:0.8300  
##  Max.   :9.920   Max.   :0.9700

sd(admission_data$CGPA)

## [1] 0.5963171

sd(admission_data$Chance.of.Admit)

## [1] 0.1426093

The CGPA has an average of 8.58, with a median of 8.56 and a minimum of 6.8 and maximum of 9.92. The standard deviation is 0.60, which indicated that most of the CGPA scores are accumulated around the mean. For the Chance of Admission, the average is 0.72, the median is 0.7217, the maximum is 0.97 and the minimum is 0.34. The standard deviation is 0.14. These values suggest that most students fall within the middle range likelihood of gaining admission into graduate school.

Model 1

# Histogram for CGPA
hist(admission_data$CGPA, 
     breaks= 12, 
     col="plum", 
     main="Distribution of CGPA", 
     xlab="CGPA", 
     border="black",
     prob=TRUE)
lines(density(admission_data$CGPA), col="red", lwd=2)

# Histogram for Chance of Admission
hist(admission_data$Chance.of.Admit, 
     breaks=10, 
     col="lightgreen", 
     main="Distribution of Chance of Admission", 
     xlab="Chance of Admission", 
     border="black",
     prob=TRUE)
lines(density(admission_data$Chance.of.Admit), col="blue", lwd=2)

The histogram of CGPA closely follows the normal curve which means that most students have a CGPA around the average 8.58 (median= 8.56). The spread is compact since most of the students fall within the first quartile 8.13 and the third quartile 9.04. The compact structure suggests that most students have a CGPA around the mean. On the other hand, the histogram for Chance of Admission is slightly right-skewed. This means that there is a small number of students who have a very low chance of gaining admission. Most of the students are accumulated around the median of 0.72. This means they have a moderate to high chance of admission into graduate programs.

Model 2

# Histogram & Fitted Distributions for TOEFL Score 
hist(admission_data$TOEFL.Score, 
     breaks=seq(90, 120, by=2), 
     col="skyblue", 
     main="Distribution of TOEFL Score", 
     xlab="TOEFL Score", 
     border="black",
     xlim=c(90,121),
     prob=TRUE,
     xaxt="n")
x_ticks <- seq(90, 120, by=2)
axis(1, at=x_ticks, labels = x_ticks)
lines(density(admission_data$TOEFL.Score), col="red", lwd=2)

# Histogram & Fitted Distribution for Chance of Admission
hist(admission_data$Chance.of.Admit, 
     breaks=seq(0,1, by=0.05),
     xlim=c(0,1),
     col="red", 
     main="Distribution of Chance of Admission", 
     xlab="Chance of Admission", 
     border="black",
     prob=TRUE)
lines(density(admission_data$Chance.of.Admit), col="blue", lwd=2)

The histogram of TOEFL Scores displays the distribution of the predictor variable x=TOEFL Scores, these values range from 0 to 120. The data displays a normal distribution with a minimum score of 92 and a maximum score of 120. The density curve follows a similar distribution as the histogram. Out of 400 applicants, the most frequent scores are 106 and 110, the average score is 107 which highlights that average applicants are competitive. The histogram of Chance of Admission displays a slightly left-skewed distribution which suggests that most of the data points are concentrated on the right side. Chance of Admission has a greater density of over 70%.

Model 3

# Histogram & Fitted Distribution for GRE Score
hist(admission_data$GRE.Score, 
     breaks=20, 
     col="blue3", 
     main="Distribution of GRE Scores", 
     xlab="GRE Score", 
     border="black",
     prob=TRUE)
lines(density(admission_data$GRE.Score), col="red", lwd=2)

# Histogram & Fitted Distribution for Chance of Admission
hist(admission_data$Chance.of.Admit, 
     breaks=10, 
     col="red2", 
     main="Distribution of Chance of Admission", 
     xlab="Chance of Admission", 
     border="black",
     prob=TRUE)
lines(density(admission_data$Chance.of.Admit), col="blue", lwd=2)

The histogram displays the distribution of the predictor variable, GRE Scores, which ranges from a typical low of around 260 to a high of 340. The data shows a normal distribution with the minimum score being 290 and a maximum of 340. The density curve depicts a similar distribution of GRE Scores. The most common scores were from between 310-325. The histogram for COA reveals the probability distribution of applicants’ chances of being accepted. A left-skewed distribution would suggest most applicants have a higher probability of admission. Because there are higher densities over 70%, the applicants scores are generally competitive.

Model 4

# Histogram & Fitted Distribution for SOP
hist(admission_data$SOP, 
     breaks=10, 
     col="lightblue", 
     main="Distribution of SOP", 
     xlab="SOP", 
     border="black",
     prob=TRUE)
lines(density(admission_data$SOP), col="red", lwd=2)

# Histogram & Fitted Distribution for Chance of Admission
hist(admission_data$Chance.of.Admit, 
     breaks=10, 
     col="red", 
     main="Distribution of Chance of Admission", 
     xlab="Chance of Admission", 
     border="black",
     prob=TRUE)
lines(density(admission_data$Chance.of.Admit), col="blue", lwd=2)

The SOP histogram shows a smooth curve, however it is slightly left-skewed with the highest points being between 3.0 and 4.0. Between 1.0 and 3.0 there is also a slightly steeper slope, meaning that there is low variability with most values falling close to the mean (3.4) and median (3.5), with few outliers. The chance of admission histogram is also slightly left-skewed, however it does not have such a smooth curve. Instead, it goes from a steep slope to a flat slope and back again, suggesting that there are central peaks with gradual spread. Comparatively, the chance of admission histogram has a steeper slope than the SOP histogram, implying that chance of admission has more data points concentrated around a specific value (0.73).

Model 1

# Quantile-Quantile plots of CGAP & Chance of Admission
qqnorm(admission_data$CGPA, main = "Q-Q Plot CGPA", ylab= "Count", xlab="Quantiles")
qqline(admission_data$CGPA, col="red")

qqnorm(admission_data$Chance.of.Admit, main = "Q-Q Plot Chance of Admission", ylab = "Count", xlab="Quantiles")
qqline(admission_data$Chance.of.Admit, col="red")

In the Q-Q plot of CGPA, the points are closely aligned to the normal distribution, but they begin to drift at the end of both tails. This suggests that there are a few outliers that may have a very high or very low CGPA. On the other hand, the Q-Q plot for Chance of Admission also closely follows the normal distribution line, with fewer points that drift off (there is one extreme outlier at the bottom).

# Boxplot of CGPA & Chance of Admission
boxplot(admission_data$CGPA, main="Boxplot of CGPA",col = "plum", border = "black", xlab=" CGPA", ylab="Count")

boxplot(admission_data$Chance.of.Admit, main="Boxplot of Chance of Admission", col="lightgreen", border = "black", xlab="Chance of Admission", ylab="Count")

The boxplot of CGPA shows that it is slightly skewed to the right, and the median line is closer to the bottom of the box. There are no major outliers and the range is spread over a moderate distance which once again confirms that the values are mostly centered around the median. The boxplot of Chance of Admission, also shows a mild right skew and the median line is closer to the bottom. There is one outlier at the bottom of the box, which suggests at least one with a very low chance of admission.

Model 2

# Quantile-Quantile plot for TOEFL Score
qqnorm(admission_data$TOEFL.Score,
       main = "QQ Plot of TOEFL Scores",
       xlab = "Quantiles",
       ylab = "Scores")
qqline(admission_data$TOEFL.Score, col="blue")

# Quantile-Quantile plot of Chance of Admit
qqnorm(admission_data$Chance.of.Admit,
       main = "QQ Plot of Chance of Admission",
       xlab = "Quantiles",
       ylab = "Count")
qqline(admission_data$Chance.of.Admit, col="blue")

The QQ Plot for TOEFL Scores shows that the majority of data points are normally distributed around the central region of scores from 100 to 115. The QQ Plot also indicates some deviations in the data points above the QQ line in quantiles (-3,-1), these scores are higher than expected of a normal distribution. The QQ Plot for Chance of Admission shows that the majority of data points are normally distributed around the central region between 60 to 90 percent. The QQ Plot also indicates some deviations in the data points below the QQ line in quantiles (-3,-1), the data has lower percentage values than expected of a normal distribution.

# Boxplot of TOEFL Score
boxplot(admission_data$TOEFL.Score, 
        main="Boxplot of TOEFL Score",
        xlab="TOEFL",
        ylab="Scores",
        col="skyblue",
        border="black",
        ylim=c(90,120))

# Boxplot of Chance of Admission
boxplot(admission_data$Chance.of.Admit, 
        main="Boxplot of Chance of Admission",
        xlab="Chance of Admission",
        ylab="Count",
        col="red",
        border="black",
        ylim=c(0,1))

The boxplot for TOEFL Scores indicates a median score of 107. The first quantile score of 103 suggests that 25% of scores are below this value. The third quantile score of 112 suggests that 75% of scores are below this value. The Interquartile Range is 9 which indicates some variability however, the data points are relatively close together and clustered around the median. The boxplot for Chance of Admission indicates a median value of 73%. The first quantile value of 64% suggests that 25% of students have a chance of admission lower than this value. The third quantile value of 83% suggests that 75% of students have a chance of admission below this value. The Interquartile Range is 19% which indicates a higher variability of the data points compared to the median.

Model 3

# Quantile-Quantile plots of GRE Score & Chance of Admission
qqnorm(admission_data$GRE.Score, main = "QQ Plot of GRE Scores",
       xlab = "Quantiles",
       ylab = "Scores")
qqline(admission_data$GRE.Score, col="blue3")

qqnorm(admission_data$Chance.of.Admit, main = "QQ Plot of Chance of Admission",
       xlab = "Quantiles",
       ylab = "Count")
qqline(admission_data$Chance.of.Admit, col="blue3")

The QQ plot for GRE scores would reveal how well the data aligns with a normal distribution. GRE scores typically cluster within a central range, like 300–320, with deviations at the extremes (lower and higher quantiles). Minor deviations might appear, suggesting scores higher or lower than expected, but a normal distribution fit would validate that GRE scores are a consistent metric. The QQ plot for COA would similarly indicate whether admission chances are normally distributed. Central values around 60-90% would align well on the QQ line, with deviations below for lower admission chances. These outliers suggest that some students have chances of admission lower than typical for their GRE score range.

# Boxplot of GRE Score & Chance of Admission
boxplot(admission_data$GRE.Score, main = "Boxplot of GRE Scores", xlab="GRE", ylab=" Scores", col="skyblue3")

boxplot(admission_data$Chance.of.Admit, main = "Boxplot of Chance of Admission", xlab="Chance of Admission", ylab="Count", col="green3")

The GRE boxplot displays the median GRE score (often around 310-320 in competitive programs). The first quantile would capture the lower 25% (e.g., scores below ~305), while the third quantile might be higher, around 330. The interquartile range (IQR) would reveal score spread, indicating GRE scores are relatively clustered but with some variability. The COA boxplot shows the distribution of admission chances. With a median value around 70%, we see that the central tendency aligns with competitive admission rates. The IQR here might be broader due to variability in other factors affecting admissions, showing that even strong GRE scores don’t guarantee high chances of admission.

Model 4

# Quantile Plots for SOP
qqnorm(admission_data$SOP, main= "QQ Plot of SOP",
       xlab = "Quantiles",
       ylab = "Values")
qqline(admission_data$SOP, col="blue2")

#Quantile Plots for Chance of Admission
qqnorm(admission_data$Chance.of.Admit, main = "QQ Plot of Chance of Admission",
       xlab = "Quantiles",
       ylab = "Count")
qqline(admission_data$Chance.of.Admit, col="blue2")

First it must be observed that the SOP dataset only deals in whole and half numbers so the graph visually looks different than the chance of admission Q-Q plot. The SOP Q-Q plot has more points left of the reference line, again suggesting that the data is left-skewed. The chance of admission Q-Q plot relatively follows the normal distribution line, meaning the distributions are similar and the data is well-approximated by the theoretical distribution. The only exception to this are the few points that drift off at the top and bottom, suggesting a couple of outliers in the dataset.

# Boxplot for SOP
boxplot(admission_data$SOP, main="Boxplot of SOP", xlab="SOP", ylab=" Values", col="lightblue")

# Boxplot for Chance of Admission
boxplot(admission_data$Chance.of.Admit, main="Boxplot of Chance of Admission", xlab="Chance of Admission", ylab="Count", col="red")

The shape of the SOP box falls between about 2.5 and 4.0 meaning that is where the majority of the data is distributed. The box sits in the middle of the graph and the median line is closer to the upper half of the box, suggesting that the data is slightly left-skewed. On the contrary, the chance of admission box plot has a median line that is very close to the middle, meaning the data is symmetrically distributed. The chance of admission box plot also shows that this data has an outlier at the bottom of the lower whisker.

Model 1

# Scatterplot of CGPA vs Chance of Admission
scatterplot (admission_data$CGPA, admission_data$Chance.of.Admit,
main = "Scatter Plot of CGPA vs Chance of Admission",
xlab = "CGPA",
ylab = "Chance of Admission",
col="plum3")

There is a positive linear relationship between CGPA and Chance of Admit, meaning that when CGPA increases the chances of gaining admission into Graduate school increases. Above 9.0 the points are closer to the trend line, which means there is a more consistent relationship between high CGPA and admission chances. However, on points below 8.0, the points are not as close to the trend line which indicates there is more variability in students with lower CGPA. This tells us that when we have a higher CGPA, it is a stronger predictor of chance of admission than if we were to have a lower CGPA (which may be more strongly influenced by other factors). There is also an outlier at 0.34.

Model 2

# Scatterplot of TOEFL Score vs Chance of Admit
scatterplot(admission_data$TOEFL.Score, admission_data$Chance.of.Admit,
     main = "Scatter Plot of TOEFL Score vs Chance of Admission",
     xlab = "TOEFL Score",
     ylab = "Chance of Admission",
     col="steelblue1")

The scatterplot visually shows that the median TOEFL Score of 107 aligns with the Chance of Admission median percentage of 73%. There is a positive relationship between TOEFL Scores and Chance of Admission. As TOEFL Scores increase, the Chance of Admission increases as well. There is a tight cluster of data points around the line however, we can see some dispersion of the data points and outliers that reveal that some students have a high TOEFL score but have low chances of admission.This reveals that there may be other factors that play a role in the admissions decision process besides TOEFL scores. Since there are outliers there may be a need for a transformation because outliers can have an impact on regression results.

Model 3

# Scatterplot of GRE Score vs Chance of Admit
scatterplot(admission_data$GRE.Score, admission_data$Chance.of.Admit,
     main = "Scatter Plot of GRE Score vs Chance of Admission",
     xlab = "GRE Score",
     ylab = "Chance of Admission",
     col="skyblue3")

The scatterplot matrix for GRE and COA should confirm the positive correlation trend, with points moving from lower-left to upper-right, suggesting a robust relationship. Tight clustering along the line supports GRE as a predictor, though some dispersed points reveal cases where high GRE scores do not correlate with higher admission chances, possibly indicating other influential factors in the admissions process. The presence of outliers might suggest a transformation to linearity could help to refine the model if outliers significantly affect the regression results.

Model 4

# Scatterplot of SOP vs Chance of Admit
scatterplot(admission_data$SOP, admission_data$Chance.of.Admit,
     main = "Scatter Plot of SOP vs Chance of Admission",
     xlab = "SOP",
     ylab = "Chance of Admission",
     col="lightblue3")

Although slightly weak, there is a positive linear relationship between the data found in the SOP scatterplot and the chance of admission scatterplot. This positive relationship suggests that the data moves with each other, meaning as the SOP score increases, so does the chance of admission. However both scatterplots have quite a few outliers, with the SOP scatterplot having significantly more outliers. This could possibly suggest that they are not as heavily correlated or reliable on one another. Both scatterplots and the curves that they produced still imply that the data is slightly left-skewed. Because of this skew, it may be helpful to apply a logarithmic or square root transformation. This could possibly make the distribution more symmetric and improve the accuracy, bringing it closer to a normal distribution.